What's the difference between 'importance' and advantage? (Not their definitions, I mean, in a general sense. What's the difference when I say 'importance of something' and 'advantage of something'?

Well, something could be important in many different ways. It might not necessarily be "advantageous," in the sense of being useful to someone. But something that offers an advantage is important only in that one sense.
That's the general sense I have of the two, completely out of context.